Aircraft: LIGHT TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Initial Approach
Anomaly: ATC Issue All Types; Conflict NMAC

I was the pilot flying sitting in the left seat. The flight was uneventful until we arrived in the terminal area at our arrival airport. The weather was clear. A beautiful Saturday mid afternoon; which meant a fair amount of VFR traffic in the traffic pattern there at BFI. With the airport in sight; we were cleared for the visual approach to Runway 31L. After checking in with the Tower and being cleared to land; we were given traffic advisories about some VFR helicopter traffic in the pattern. Tower advised that they were close and that the helicopter traffic had us in sight. We had the map on the MFD scrolled in so that we could see the traffic in our area. At approximately 700 AGL the TCAS began with a 'TA'. Almost immediately we received an 'RA'. At a quick glance; I recall seeing the conflicting traffic below us by 400 FT and we were descending. By now the TCAS is in full tenor singing 'CLIMB! CLIMB NOW'!!! So...we climb. I called out 'GO-AROUND' and both my Co-Captain and I executed the missed approach. All along the Tower Controller was so busy with his duties and the traffic that he sounded like an auctioneer. My Co-Captain had to jump in and announce that we had gone missed. After what seemed like a bit of a pause; my partner asked the Tower for missed approach instructions. The Tower Controller responds with...'Fly the published missed.' And he said it as if he was annoyed. Annoyed with what? The fact we didn't trust his traffic advisories or the fact that we had the nerve to ask for missed approach instructions. Either way; I didn't think visual approaches came with a published missed approach procedure. Nonetheless; since we did in fact have the ILS procedure up and available; we started to maneuver in the proper direction. As we made our initial turn to the left toward the LOFAL intersection; heading approx 280 and climbing through approx 2;000 FT MSL; the Tower Controller is yelling at us... 'XXX descend; descend now! Traffic at your 10 o'clock!' Startled; as I pulled the power back and pushed the nose over; I look to my left and see a B-737 climbing. He's above us; but not by much. Also our horizontal separation isn't by much either. At the beginning of this fiasco; I had to climb to avoid an in-flight collision. Now; just a mere 30 seconds later; I have to descend to accomplish the same. The Tower Controller told us to descend and maintain 1;500 FT. So; we descend. After what seemed like awhile listening to the Tower Controller going back to his auctioneer style of controlling the traffic we left behind; my partner again jumps in and asked if we should go to departure? 'YES! YES PLEASE!' was the response. So... we contact Departure Control. Checking in with Departure was uneventful. No hype or hysteria here. The Controller acknowledged our heading and altitude. Moments later he gave us a clearance to climb to 4;000 FT and vector for another visual approach into BFI. We briefly mentioned to the Departure Controller the nature of our missed approach out of BFI. We told him that our concern was another repeat of the same scenario; and the primary concern was the aggressive helicopters that fly a very tight pattern and close proximity to other arriving traffic. He mentioned that he would pass this along to the Tower at BFI. The rest of the flight was routine except my contempt for the Tower Controller and his misfit helicopter pilots. All the more vigilant; we flew another visual approach to 31L. Checking in with what seemed like a different Tower Controller; we were cleared to land. And so... we landed. Please review the missed approach procedure for the ILS 31L into BFI. Please note the left hand turn that is required to head toward LOFAL intersection. That heading takes you right across the departure path of SEA. Also; advise the BFI Tower that a sensitive TCAS; along with an errant/aggressive helicopter flying technique isn't a good combination. The runways at BFI are very closely spaced. Until that day I was unaware of what seems to be pretty intensive helicopter training at BFI; in very small aircraft that are not easy to pick out in the ground clutter. Perhaps more separation should be provided by the controllers during sequencing because even if visual contact is made; a TCAS RA must be complied with. A fractional crew executed a go-around while approaching Runway 31L at BFI after a TCAS RA from a small helicopter landing on Runway 31R. Aircraft comes in conflict with a B737 departing nearby SEA while on the Tower issued missed approach procedure.